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DEVELOPMENT OF DESIGN EQUATIONS AND GUIDELINES FOR FIBER-

REINFORCED POLYMER (FRP) COMPOSITE CHANNEL SECTIONS

Final report submitted to

Creative Pultrusions, Inc., Alum Bank, PA

By

Pizhong Qiao (Chiao), Ph.D., P.E.

Associate Professor of Structural Engineering and Advanced Materials

Department of Civil Engineering, The University of Akron, Akron, OH 44325-3905

Phone: (330) 972-5226; Fax: (330) 972-6020; Email: [email protected]

July 15, 2003

2

TABLE OF CONTENTS

Title Page

Cover Page……………………………………………………………………………………1

Table of Contents……………………………………………………………………………2

List of Figures……………………………………………………………………………........3

List of Tables…………………………………………………………………………….........4

Introduction…………………………………………………………………………………5

Objectives……………………………………………………………………………………...5

Methodology…………………………………………………………………………………6

FRP channel sections…………………………………………………………………7

Panel material properties……………………………………………………………8

Local buckling of FRP channels……………………………………………………10

Lateral buckling of FRP channels…………………………………………………15

Master design curves………………………………………………………………30

Design Guideline……………………………………………………………………………34

Conclusions…………………………………………………………………………………34

Acknowledgements…………………………………………………………………………35

References……………………………………………………………………………………35

3

LIST OF FIGURES

Title Page

Figure 1 FRP channel shapes………………………………………………………………...7

Figure 2 Modeling of local buckling of FRP channel shapes……………………………....9

Figure 3 Local buckling deformed shapes (First Mode) for channel 10"x2-3/4"x1/2"

(C10x3) with Length = 50.0 ft………………………………………………………14

Figure 4 Cantilever configuration of FRP channel beam…………………………….......21

Figure 5 Load applications at the cantilever tip through the shear center……………...21

Figure 6 Buckled channel C4x1 beam (L = 11.0 ft.)……………………………………….22

Figure 7 Buckled channel C6x2-A beam (L = 11.0 ft.)…………………………………….23

Figure 8 Buckled channel C6x2-B beam (L = 11.0 ft.)………………………………….....24

Figure 9 FE simulation of buckled C4x1 beam……………………………………………25

Figure 10 FE simulation of buckled C6x2-A beam………………………………………..25

Figure 11 FE simulation of buckled C6x2-B beam………………………………………..26

Figure 12 FE simulation of buckled C8x2 beam…………………………………………26

Figure 13 FE simulation of buckled C10x3 beam…………………………………………27

Figure 14 Comparison of lateral buckling for C4x1……………………………………....28

Figure 15 Comparison of lateral buckling for C6x2-A……………………………………28

Figure 16 Comparison of lateral buckling for C6x2-B…………………………………....29

Figure 17 Comparison of lateral buckling for C8x2………………………………………29

Figure 18 Comparison of lateral buckling for C10x3……………………………………..30

Figure 19 Design curve for C4x1 (Moment capacity vs. unbraced length relationship)..31





4

LIST OF TABLES

Title Page

Table 1 Panel engineering properties of five FRP channel shapes…………………...……9

Table 2 Local buckling analysis of five FRP channel shapes……………………………..13

Table 3 Comparisons of FE model and proposed explicit design equations for local

buckling of channel sections………………………………………………………...14

Table 4 Comparison of lateral buckling loads of channel section (C4x1)………………..17

Table 5 Comparison of lateral buckling loads of channel section (C6x2-A)…………….18

Table 6 Comparison of lateral buckling loads of channel section (C6x2-B)……………..19

Table 7 Comparison of lateral buckling loads of channel section (C8x2)………………..20

Table 8 Comparison of lateral buckling loads of channel section (C10x3)………………20

5

INTRODUCTION

In recent years, there has been a considerable increase in the use of pultruded fiber-

reinforced plastic (FRP) shapes (e.g., beams, columns, and cellular deck panels) in structural

applications. FRP beams have shown to provide efficient and economical applications in

bridges and piers, retaining walls, airport facilities, storage structures exposed to salts and

chemicals, and others. FRP materials are lightweight, noncorrosive, nonmagnetic, and

nonconductive. In addition, they exhibit excellent energy absorption characteristics,

suitable for seismic response; high strength, fatigue life, and durability; competitive costs

based on load-capacity per unit weight; and ease of handling, transportation, and

installation. Also, monitoring sensors, such as fiber optics, can be easily integrated into

FRP composites during manufacturing. Moreover, FRP composites offer the inherent

ability to alleviate or eliminate the following four construction related problems adversely

contributing to transportation deterioration worldwide (Head 1996): corrosion of steel, high

labor costs, energy consumption and environmental pollution, and devastating effects of

earthquakes.

Even though substantial research on FRP shapes has been reported in the literature, there

is a need to translate the useful research results into practice. The lack of design procedures

and equations for FRP shapes presents a problem to builders, government officials,

administrators and practicing engineers. Several studies have been conducted on

characterization, analysis and design of FRP shapes (Davalos, Barbero, and Qiao 2002), and

FRP sections studied are mostly in double-symmetric configurations (e.g., I and box sections).

There are no design guidelines and analysis available for FRP channel sections. In this report,

a combined experimental and analytical study of FRP channel sections is conducted, and a

design guideline for analysis of FRP channel shapes is developed.

OBJECTIVES

Singly symmetric sections (e.g., Channel sections) are used frequently in construction.

Due to the relatively low stiffness of polymer (e.g., Polyester or vinylester resins are

6

commonly used in pultruded products) and thin-walled sectional geometry of FRP shapes,

problems with large deformation and local and global buckling are common phenomena in

current structural design and analysis (Qiao et al. 1999). In particular, design equations and

procedures for stability analysis of FRP channel sections are not available. In consideration

of the current design needs for FRP channel sections, we aim to accomplish the following

objectives in this study:

1. Develop local buckling design equations and provide design recommendations for

improving local buckling capacity of the channel sections by lateral supports or

restraint supports;

2. Experimentally characterize lateral buckling of channel sections and develop global

buckling design equation;

3. Develop master design charts for buckling of FRP channel sections, in which the

moment capacity vs. unbraced length design relationship (curve) will be provided by

considering both local and global buckling behaviors.

4. Provide a design summary and procedure for FRP channel sections.

In this study, the panel material properties of several common FRP channel sections are

first provided, based on manufacturer information (e.g., lay-up and material properties) and

micro/macromechanics model (Davalos et al. 1996). Using a discrete plate analysis of flange

and web panels, the local buckling of channel shapes is then studied, and related explicit

design equations are developed. Combined experimental, numerical and analytical study of

global (lateral) buckling of FRP channels is conducted, and a simplified equation is proposed.

Finally, master curves for stability analysis of FRP channels and design procedures are

summarized.

METHODOLOGY

This section is concerned with the development of design equations for stability analysis

of FRP channel sections. The design equations are developed based on combined

experimental, analytical and numerical study of five FRP channel sections, which are

7

representative of the shapes currently used in practice. In this report, the design data is

provided for these five representative “commercial” channel FRP sections available from

Creative Pultrusions, Inc., Alum Bank, PA.

FRP Channel Sections

Design equations are developed in this study based on design parameters tabulated for five

representative channel section currently produced by Creative Pultrusions, Inc. The following

design parameters are considered: panel stiffness, flange and web local buckling loads, and

lateral buckling loads. The five beams studies include the following shapes (Figure 1,

dimensions of height (h) x width (w) x thickness (t)): Channel 4”x1-1/8”x1/4” (C4x1),

Channel 6”x1-5/8”x1/4” (C6x2-A), Channel 6”x1-11/16”x3/8” (C6x2-B), Channel 8”x2-

3/16”/3/8” (C8x2), and Channel 10”x2-3/4”x1/2” (C10x3). All the five channel sections are

analyzed, and the developed analytical solutions and design formulas are compared with the

commercial finite element modeling using ANSYS.

Figure 1 FRP channel shapes

w

h

t

t

e x x

Centroid

y

y

Shear center

8

Panel Material Properties

Even though the panels of FRP structural beam are not manufactured by hand lay-up, the

pultruded panels can be simulated as a laminated structure. For most pultruded FRP sections,

the lay-up of a panel is usually balanced symmetric; the panel stiffness properties are

orthotropic and can be obtained by theoretical predictions of micro/macromechanics (Davalos

et al. 1996).

Most pultruded FRP sections such as channel (C) beams consist typically of arrangements

of flat panels (see Figure 1) and mainly include the following three types of layers (Davalos et

al. 1996): (1) Continuous Strand Mats (CSM); (2) +/-α Stitched Fabrics (SF); and (3) rovings

or unidirectional fiber bundles. Usually, the reinforcement used is E-glass fibers, and the

resin or matrix is either vinylester or polyester. Each layer is modeled as a homogeneous,

linearly elastic, and generally orthotropic material. Based on the fiber volume fraction and

the manufacturer’s information, the ply stiffness can be computed from micromechanics

models for composites with periodic microstructure (Luciano and Barbero 1994). Then, the

stiffness of a panel can be computed from classical lamination theory (CLT) (Jones 1976). In

CLT, the engineering properties (Ex, Ey, ν xy, and Gxy) of the panel are computed by

assembling the transformed stiffness coefficients into the extensional stiffness matrix [A].

The engineering properties of the pultruded panel are then computed as (Davalos et al. 1996):

E t E t G tx y xy xy= = = − =1 1 111 22 12 11 66/ ( ), / ( ), / , / ( )α α ν α α α (1)

where t is the panel thickness; [α ] is the compliance matrix, which is the inverse of the

extensional stiffness matrix [A]. By using micro/macromechanics model (Davalos et al.

1996) and FRPBEAM program (Qiao et al. 1994), the panel stiffness properties of E-

glass/polyester composites are computed for five channel sections and shown in Table 1. As

introduced in the following sections, explicit equations, which can be applied in engineering

design, for the computation of beam local/global buckling loads are developed in terms of

panel stiffness and strength properties.

9

Table 1 Panel engineering properties of five FRP channel shapes

Section Ex

(x106 psi)

Ey

(x106 psi)

Gxy

(x106 psi)

vxy vyx

Channel 4"x1-1/8"x1/4" (C4x1) 2.857 1.633 0.568 0.373 0.213

Channel 6"x1-5/8"x1/4" (C6x2-A) 3.728 1.843 0.651 0.359 0.177

Channel 6"x1-11/16"x3/8" (C6x2-B) 3.292 1.616 0.570 0.360 0.177

Channel 8"x2-3/16"x3/8" (C8x2) 3.292 1.616 0.570 0.360 0.177

Channel 10"x2-3/4"x1/2" (C10x3) 3.704 1.709 0.606 0.355 0.164

Figure 2 Modeling of local buckling of FRP channel shapes

x

x

y

y

Flange

Web

Free edge

Restraint from web (a) discrete plate of flange

(b) discrete plate of web

a

h

w Ncr Ncr

Ncr Ncr

a b

b

Restraint from flange

Restraint from flange

10

Local Buckling of FRP Channels

A variational formulation of Ritz method (Qiao and Zou 2002; Qiao and Zou 2003) is

used to establish an eigenvalue problem, and the flange and web critical local buckling

coefficients are determined. In the local buckling analysis, the panels of FRP channel shapes

are simulated as discrete laminated plates or panels (Figure 2), and the effects of restraint at

the flange-web connection are considered. The flange of pultruded FRP channel sections is

modeled as a discrete panel with elastic restraint at one unloaded edge and free at the other

unloaded edge (restrained-free (RF) condition) and subjected to uniform distributed axial in-

plane force along simply supported loaded edges (Figure 2a); whereas for the web, the panel

is modeled as a discrete plate with equal elastic restraints at two unloaded edges (restrained-

restrained (RR) condition) (Figure 2b).

For the flange panels under compression, the formula of plate local buckling strength,

(Nx)cr, with elastically restrained at one unloaded edge and free at the other (Figure 2a) is

given as (Qiao and Zou 2003)

( ) [

]xy

yxxyyf

crx

G

EEvEb

tN

)364(40

)10156()2(49.15)32(20)10156(12

2

222

3

ξξ

ξξξξξξ

+++

+++++−++

=

(2)

where, Ex, Ey, Gxy and vxy are the material properties of the flange panel, and they are given in

Table 1; t and bf are the thickness and width of the flange panel; ξ is the coefficient of

restraint for flange-web connection and is given as

)2()()(61

1

2

2

2xyxyyyx

xyf

wf

w

GvEEEG

bbb

b

++−

=

π

ξ (3)

where bw is the width of the web panel (Figure 2). The critical aspect ratio (γ = a/b, where a

is the length of the panel) of the flange panel is defined as

41

2

2

)2()91810(

1287.1

+++

=y

xcr E

Emξ

ξξξγ (4)

11

where m is the number of buckling half waves.

For the web panel under compression, the formula of plate local buckling strength, (Nx)cr,

with equal elastic restraints (RR conditions) at the unloaded edges or web-flange connections

(Figure 2b) is given as (Qiao and Zou 2002)

)]2(871.1[2)(1

3

1

22

3

xyxyyyxw

crx GvEEEb

tN ++=ττ

ττ (5)

where 321 ,, τττ are functions of the elastic restraint coefficient ξ , and defined as

195.25 ,176 ,11131 23

22

21 ++=++=++= ξξτξξτξξτ

and ξ is the coefficient of restraint contributed by the flange and expressed as:

xy

xyxyyyx

w

f

f

w

GGvEEE

bb

bb

)2(

61

)(2

22

2

++

−

=π

ρξ (6)

and )/cosh()/sinh(3/

1)/()/(cosh321)(

22

2fwfwfw

fwfw

f

w

bbbbbbbbbb

bb

πππππ

πρ

+

++= . The elastic restraint coefficient ξ

= 0 corresponds to clamped restraints at the flange-web connection; while ∞=ξ to simply

supported restraints at the flange-web connection. For simplified design of the web panels

under compression, two unloaded edges can be approximately simulated as simply supported

(Figure 2b) since the adjacent flanges are relatively free to rotate and their restrain to the web

is minimal and can be neglected. Then, the formula of plate local buckling strength, (Nx)cr,

with simply supported conditions (SS condition) at the unloaded edges or flange-web

connections is given as (Qiao et al. 2001)

)}2({6

)( 2

32

xyxyyyxw

crx GvEEEb

tN ++=π (7)

The critical aspect ratio of the web panel is defined as

4

y

xcr E

Em=γ (8)

where m is the number of buckling half waves.

12

The explicit formulas of respective local buckling loads of flange and web panels are

given in Eqs. (2) and (7), respectively, and the lower value obtained from the Eqs. (2) and (7)

controls the local buckling loads of the channel shapes under compression. The explicit

formulas for the critical aspect ratio (γ = a/b, where a = length and b = width) are also given

in Eqs. (4) and (8), respectively, for flange and web panels, of which the desirable locations of

restraint supports or bracings can be obtained. Based on the critical aspect ratios for local

buckling, the number and locations of restraint (or lateral) supports can be recommended and

properly designed.

Design procedures for local buckling: Based on the formulas presented above, the

following design procedures are recommended for local buckling design of channel shapes:

a. Compute the critical local buckling loads ((Nx)cr) of flange and web panels,

respectively, using Eqs. (2) and (7).

b. Compare the local buckling loads obtained in (a); the lower value ((Nx)cr) of Eqs. (2)

and (7) will control the local buckling of the channel sections.

c. Compute the axial compressive local buckling load (Pcr) of the channel section using

the control panel local buckling strength value ((Nx)cr) evaluated in (b) as

)2()()( whNP crxaxialcr += (9)

where h and w are the height and width of the channel section, respectively (see

Figures 1 and 2).

d. Identify the control mode (i.e., which panel will first buckle?) based on the conclusion

in (b). If ((Nx)cr)flange < ((Nx)cr)web, then the flange controls the local buckling of the

channel section, and use Eq. (4) to compute the critical aspect ratio (γcr); otherwise, the

web controls the local buckling of the channel, and use Eq. (8) to compute the critical

aspect ratio.

e. Use critical aspect ratio identified in (d) to obtain the locations of restraints or lateral

bracings so that the local buckling capacity of the chancel can be improved.

The local buckling parameters (e.g., (Nx)cr, ξ, and γcr) for five representative channel

sections (see their panel material properties in Table 1) are provided in Table 2. As shown in

Table 2, for all the given five channel sections, the flange gives a lower ((Nx)cr) value

13

compared to the web; thus the flange will buckle first and control the local buckling of the

selected channel sections.

Table 2 Local buckling analysis of five FRP channel shapes

Section

(Nx)cr

Flange

(lb/in)

Eq. (2)

ξ

Eq. (3)

γcr

Flange

Eq. (4)

m =1

(Nx)cr

Web

(lb/in)

Eq. (7)

γcr

Web

Eq. (8)

m = 1

(Pcr)axial

(lb)

Eq. (9)

Channel 4"x1-1/8"x1/4"

(C4x1) 7013 3.02 0.526 7133 1.15 43831


(C6x2-A) 3852 2.12 0.502 3864 1.19 35631

Channel 6"x1-11/16"x3/8"

(C6x2-B) 10556 3.96 0.579 11044 1.20 98963

Channel 8"x2-3/16"x3/8"

(C8x2) 6282 2.43 0.520 6389 1.20 77740

Channel 10"x2-3/4"x1/2"

(C10x3) 10017 2.89 0.550 11004 1.21 155264

Finite element modeling: To validate the proposed design equation, the commercial finite

element package ANSYS is employed for modeling of the local buckling of channel sections

using Mindlin eight-node isoparametric layered shell element (SHELL 99). A reasonable

correlation between the FE model and explicit design equation is achieved (see Table 3); a

maximum of difference of 11.4% is observed for Channel 6"x1-11/16"x3/8" (C6x2-B). An

illustration of local deformed shape of buckled channel section (C10x3) is shown in Figure 3.

14

Table 3 Comparisons of FE model and proposed explicit design equations for local

buckling of channel sections

Section (Nx)cr (FE) (lb/in)

(Nx)cr (Design) (lb/in)

(Nx)crFE/(Nx)cr

Design


(C4x1)

7604

7013

1.084


(C6x2-A)

3668

3852

0.952

Channel 6"x1-11/16"x3/8"

(C6x2-B)

11757

10556

1.114

Channel 8"x2-3/16"x3/8"

(C8x2)

6202

6282

0.987

Channel 10"x2-3/4"x1/2"

(C10x3)

10983

10017

1.096

Figure 3 Local buckling deformed shapes (First Mode) for channel 10"x2-3/4"x1/2"

(C10x3) with Length = 50.0 ft.

15

Lateral Buckling of FRP Channels

For lateral buckling of FRP Channels, there are no explicit formulas available for

prediction of critical buckling load or moment, due to unsymmetric nature of the cross section

and complexity of the problem. In this study, some available explicit formulas, which are

used for design analysis of FRP I-beams (Pandey et al. 1995), are adapted for prediction of

lateral buckling capacity of channel sections. Since two back-to-back channels have similar

behavior of I-section, the lateral buckling behavior of a single channel may be similar to the

one of I-section; however, the applied load in the channel section must be acted at the shear

center of the channel section (see Figure 1). For uniform FRP channel section (i.e., both the

web and flanges have the same material properties and thickness), the shear center (e in

Figure 1) can be simply defined as (Boresi and Schmidt 2003):

whwor

bbb

efw

f

63

63 22

+=

+= (10)

where bf (= w) and bw (= h) are the widths of flange and web, respectively (Figure 1).

For long-span FRP channel beams without lateral supports and with relatively large

slenderness ratios, the lateral buckling is prone to happen. Based on Vlasov’s theory (Pandey

et al. 1995), a simplified engineering equation for lateral buckling of an “I” section is adapted

for prediction of lateral buckling of channel section. The lateral buckling of channel section

under a tip load (cantilever beam) through the shear center is approximated as

2L

JGIEP yyx

cr γ= (11)

where

+

+++= 2)10(

)3)(13(3.19108.5κ

κκκ

γ

wwIJGL2

=κ

JGG t b G t bxy f f f xy w w w= +

23 3

3 3( ) ( )

16

IE t b b E t b E t b

wwx f f w f x f f f x w w w= + +

( ) ( ) ( )2 3 3 3 3 3

24 36 144

and Iyy is the moment of inertial of the channel section along the weak axis.

Based on Eq. (11), the critical lateral buckling loads applied through the shear center for

cantilever channels are given in Tables 4 and 8 (see Pcr under “design” columns). Eq. (11) is

used in this study as a design formula for lateral buckling of FRP channel sections. The

global (or lateral) buckling loads predicted by Eq. (11) correspond to the critical loads passing

through the shear center of the channel section and causing the sideway and rotation of the

beam without distortion of cross section (flexural-torsion or lateral buckling). To validate the

accuracy of Eq. (11), both experimental testing and finite element numerical simulation are

conducted.

Experimental characterization of lateral buckling of FRP cantilever channel beams: In

this study, three geometries of FRP channel beams, which were manufactured by the

pultrusion process and provided by Creative Pultrusions, Inc., Alum Bank, PA, were tested to

evaluate their lateral buckling responses. The three channel sections consist of (1) Channel

4"x1-1/8"x1/4" (C4x1); (2) Channel 6"x1-5/8"x1/4" (C6x2-A); and (3) Channel 6"x1-

11/16"x3/8" (C6x2-B). The clamped-end of the beams was achieved using wood clamp and

insert case pressured by the Baldwin machine (Figure 4). A piece of aluminum angle with

groove was attached to the channel beam tip, and the location of loading could be adjusted so

that the load was applied through the shear center (Figure 5). Using a loading platform

(Figure 5), the loads were initially applied by sequentially adding steel plates, and as the

critical loads were being reached, incremental weights of steel plates were added until the

beam buckled. The tip load was applied through a chain attached at the shear center of the

cross section (Figure 5). One level was used to monitor the rotation of the cross section, and

the sudden sideway movement of the beam was directly observed in the experiment. The

representative buckled shapes of three channel geometries at a span length of 11.0 ft. are

shown in Figures 6 to 8, and their corresponding critical loads were obtained by summing the

weights added during the experiments. Varying span lengths for each geometry were tested;

two beam samples per geometry were evaluated, and an averaged value for each pair of beam

17

samples was considered as the experimental critical load. The measured critical buckling

loads and comparisons with analytical solutions and numerical modeling results are given in

Tables 4 to 6 (see Pcr under “experiment” columns).

Table 4 Comparison of lateral buckling loads of channel section (C4x1)

Channel )14("41"

811"4 ××× C ; shear center: "308.0=e

L (ft.) Pcr (lb)

(Design)

Pcr (lb)

(Experiment)

Pcr (lb)

(FE)

2 696.01 - 807.96

3 273.15 239.32 323.83

4 145.33 136.46 168.75

5 90.13 89.56 102.55

6 61.46 63.09 68.705

7 44.64 45.09 48.975

8 33.94 31.87 36.654

9 26.67 23.06 28.484

10 21.52 18.87 22.722

11 17.74 14.37 18.578

12 14.87 14.06 15.710

13 12.65 12.06 13.288

14 10.89 10.06 11.282

15 9.48 - 9.765

16 8.32 - 8.492

18

Table 5 Comparison of lateral buckling loads of channel section (C6x2-A)

Channel )26("41"

851"6 AC −××× ; shear center: "458.0=e

L (ft.) Pcr(lb)

(Design)

Pcr(lb)

(Experiment)

Pcr(lb)

(FE)

6 178.68 - 242.37

7 126.17 159.56 168.72

8 93.59 126.49 123.46

9 72.23 93.81 93.729

10 57.6 73.47 73.876

11 46.92 53.84 59.954

12 39.03 43.83 48.971

13 32.98 37.34 41.07

14 28.25 28.84 34.734

15 24.47 22.39 29.984

16 21.40 - 25.746

19

Table 6 Comparison of lateral buckling loads of channel section (C6x2-B)

Channel )2x6("83"

16111"6 BC −×× ; shear center: "4615.0=e

L (ft.) Pcr (lb)

(Design)

Pcr (lb)

(Experiment)

Pcr (lb)

(FE)

6 328.79 275.81 403.13

7 235.61 174.91 288.38

8 177.31 132.89 214.26

9 138.37 110.01 165.34

10 111.07 95.31 129.12

11 91.16 75.62 99.528

12 76.19 62.27 86.676

13 64.65 51.43 74.322

14 55.55 42.60 63.932

15 48.26 36.33 54.168

16 42.32 30.11 47.5

20


Channel )28("83"

1632"8 ××× C ; shear center "611.0=e

L (feet) Pcr (lb) (FE) Pcr (lb)(Design)

6 885.75 678.99

7 616.69 474.04

8 453.52 349.33

9 347 268.09

10 271.77 212.31

11 220.95 172.37

12 180.27 142.80

13 150.23 120.29

14 127.1 102.75

15 108.89 88.82

16 92.908 77.56


Channel )310("21"

432"10 ××× C ; shear center: "765.0=e

L (feet) Pcr (lb) (FE) Pcr (lb) (Design)

6 2749 2152.59

7 1946.7 1490.14

8 1432.7 1089.68

9 1090.2 830.58

10 860.7 653.88

11 686.98 528.20

12 565.5 435.70

13 470.3 365.66

14 397.18 311.36

15 339.91 268.40

16 293.63 233.83

21

Figure 4 Cantilever configuration of FRP channel beam

Figure 5 Load applications at the cantilever tip through the shear center

22

Figure 6 Buckled channel C4x1 beam (L = 11.0 ft.)

23

Figure 7 Buckled channel C6x2-A beam (L = 11.0 ft.)

24

Figure 8 Buckled channel C6x2-B beam (L = 11.0 ft.)

25

Finite element modeling: Again, to validate the accuracy of design equation (Eq. (11)), the

channel sections with various span lengths were modeled using ANSYS, and the eigenvalue

study was conduct, of which the critical loads for buckling were predicted. In the FE

modeling, the tip concentrated load was applied to the shear center through infinite rigid bar

elements. The FE results for the five given channel sections are provided in Tables 4 to 8 (see

Pcr under “FE” columns), and the FE simulations of deformed shapes under buckling are

given in Figures 9 to 13.

(a) L = 2 ft. (b) L = 16 ft.

Figure 9 FE simulation of buckled C4x1 beam

(a) L = 6 ft. (b) L = 16 ft.

Figure 10 FE simulation of buckled C6x2-A beam

26

(a) L = 6 ft. (b) L = 16 ft.

Figure 11 FE simulation of buckled C6x2-B beam

(a) L = 6 ft. (b) L = 16 ft.


27

(a) L = 6 ft. (b) L = 16 ft.


Comparisons and discussion: The comparisons of critical lateral buckling loads among

experimental, numerical (FE), and design equations (Eq. (11)) are given in Tables 4 to 8 and

graphically shown in Figures 14 to 18, respectively, for five channel sections. As expected,

the critical load decreases as the span length increases and lateral buckling becomes more

prominent. As shown in Figures 14 to 18, good correlations among design equation,

experiment, and FE results are achieved for relatively long span (e.g., L > 4 ft. for C4x1 and L

> 10 ft. for the rest of sections); while for shorter span lengths, the buckling mode is more

prone to lateral distortional instability which is not considered in the present study. This

phenomenon can also be observed in Figures 9 to 13, where the critical buckling mode shapes

are shown for the buckled channel beams with the respective short and long span lengths

using finite element modeling (ANSYS). Also for most of channel sections, the design

equation (Eq. (11)) provides a lower bound compared to FE and experimental data, except for

the case of C6x2-B; therefore, Eq. (11) can be served as a conservative design equation to

predict the lateral buckling of cantilever FRP channel shapes loaded at the shear center.

28

Length (ft.)

2 4 6 8 10 12 14 16

Pcr

(lbs

)

0

200

400

600

800

DesignExperimentalFE

Figure 14 Comparison of lateral buckling for C4x1

Length (ft.)

6 8 10 12 14 16

Pcr

(lbs

)

0

50

100

150

200

250


Figure 15 Comparison of lateral buckling for C6x2-A

29

Length (ft.)

6 8 10 12 14 16

Pcr

(lbs

)

0

100

200

300

400

500


Figure 16 Comparison of lateral buckling for C6x2-B

Length (ft.)

6 8 10 12 14 16

Pcr

(lbs

)

0

200

400

600

800

1000

FEDesign


30

Length (ft.)

6 8 10 12 14 16

Pcr

(lbs

)

0

500

1000

1500

2000

2500

3000

FEDesign


Master Design Curves

Based on the design formulas and critical loads for local and global (lateral) buckling, a

master design chart for stability of FRP channel beam is developed. The master design curve

provides the relationship between sectional moment capacity (Mcr) vs. unbraced length (L) for

a given channel section. The sectional moment capacity consists of (a) local buckling

moment which is a constant for a given FRP channel beam and suitable for short span length:

)2/()(

htINM xxcrx

cr = (12)

where Ixx is the moment of inertia about the strong axis of the channel beam (Figure 1); (Nx)cr

is the critical flange local buckling strength obtained from Eq. (2); and (b) global (lateral)

moment which is inversely proportional to the beam length and more suitable for long span

length:

LPM crcr = (13)

31

where Pcr is based on the design prediction given in Eq. (11) for a cantilever beam.

The master design charts (curves) for five representative channel sections are given in

Figures 19 to 23. The constant (horizontal) portion of the curve represents the local buckling

behavior of flange caused by the bending of the channel beam; while the remaining part

corresponds to the global (lateral) buckling. The curves shown in Figures 19 to 23 represent a

design envelop for channel sections, under which provide a safe zone for structural stability.

Length (ft.)

0 2 4 6 8 10 12 14 16

Mcr

(lbs

-in)

0

10000

20000

30000

40000

50000

6000054,495 lbs-in

Figure 19 Design curve for C4x1 (Moment capacity vs. unbraced length relationship)

32

Length (ft.)

0 2 4 6 8 10 12 14 16

Mcr

(lbs

-in)

0

10000

20000

30000

40000

50000

60000

70000

60,559 lbs-in

Figure 20 Design curve for C6x2-A (Moment capacity vs. unbraced length relationship)

Length (ft.)

0 2 4 6 8 10 12 14 16

Mcr

(lbs

-in)

0.0

2.0e+4

4.0e+4

6.0e+4

8.0e+4

1.0e+5

1.2e+5

1.4e+5

1.6e+5

1.8e+5170,216 lbs-in

Figure 21 Design curve for C6x2-B (Moment capacity vs. unbraced length relationship)

33

Length (ft.)

0 2 4 6 8 10 12 14 16

Mcr

(lbs

-in)

0.0

2.0e+4

4.0e+4

6.0e+4

8.0e+4

1.0e+5

1.2e+5

1.4e+5

121,976 lbs-in


Length (ft.)

0 2 4 6 8 10 12 14 16

Mcr

(lbs

-in)

0

1e+5

2e+5

3e+5

4e+5

5e+5442,418 lbs-in


34

DESIGN GUIDELINE

To facilitate the design of stability of FRP channel sections, a step-by-step design

guideline is recommended as follows:

1. Obtain the panel material properties of channel sections from either micro/macro

mechanics (Davalos et al. 1996) combined with the FRPBEAM program (Qiao et al.

1994) (see Table 1) or Carpet Plots (Davalos, Barbero, and Qiao 2002).

2. Use Eq. (2) and (7) to predict the local buckling strength of flange and web panels, of

which the low value controls the local buckling of the channel section.

3. Use Eq. (9) to evaluate the local buckling capacity of the channel section under axial

compression.

4. Design the locations and the number of stiffeners or bracings to enhance the local

buckling strength of channel sections using the critical aspect ratios given in Eq. (4) or

Eq. (8) (depending on the vulnerability of flange or web – the panel buckled first

control the design).

5. Predict the global (lateral) buckling of channel sections using Eq. (11).

6. Develop the master design plot for stability of channel beams based on the local (step

2) and global (step 5) buckling designs.

CONCLUSIONS

In this report, simplified design equations and procedures for stability (local and global

buckling) of FRP channel sections are developed, and the accuracy and validity of the design

equations are verified numerically by the finite element modeling and experimentally (for

global buckling only) by testing three representative channels out of 5 sections. The local

buckling of channel sections is achieved through discrete plate analyses of flange and web

panels, respectively; while the design equation for global (lateral) buckling is obtained by

adapting the existing formula used for composite “I” beam section and with condition of

applied load through the shear center of channel sections. The master design curves which

35

represent the relationship of moment capacity vs. unbraced span length of channel beams are

correspondingly developed using the local and global buckling design parameters, and they

provide a safe design envelop for the stability of channel sections.

ACKNOWLEDGEMENTS

The writer wants to thank the support and samples provided by the Creative Pultrusions,

Inc., Alum Bank, PA and Dustin Troutman, Technical Sale Manager of CP for his patience

and continuing support. The finite element modeling was performed by Guanyu Hu; while

the experimental works were conducted with assistance of Geoffrey A. Markowski. Their

help in the numerical simulation and experimental works are greatly appreciated.

REFERENCES:

Boresi, A.P. and Schmidt, R.J. 2003. Advanced Mechanics of Materials. 6th edition, John

Wiley & Sons, Inc. New York, NY.

Davalos, J.F., Salim, H.A., Qiao, P., Lopez-Anido, R. and Barbero, E.J. 1996. “Analysis and

Design of Pultruded FRP Shapes under Bending.” Composites, Part B: Engineering

Journal, 27(3-4):295-305.

Davalos, J.F., Barbero, E.J., and Qiao, P.Z. 2002. “Step-by-step Engineering Design

Equations for Fiber-reinforced Plastic Beams for Transportation Structures,” Final Report

for Research Project (RP#147), West Virginia Department of Transportation. 39 pages.

Head, P.R., 1996. “Advanced Composites in Civil Engineering - A Critical Overview at This

High Interest, Low Use Stage of Development,” Proceedings of ACMBS, M. El-Badry,

ed., Montreal, Quebec, Canada, pp. 3-15.

36

Jones, R.M., 1975. Mechanics of composite materials. Hemisphere Publishing Corporation,

New York, NY.

Luciano, R. and Barbero, E.J., 1994. "Formulae for the stiffness of composites with Periodic

microstructure," Int. J. of Solids and Structures, 31 (21), 2933.

Pandey, M.D., Kabir, M.Z., and Sherbourne, A.N. 1995. "Flexural-torsional stability of thin-

walled composite I-section beams," Composites Engineering, 5(3): 321-342.

Qiao, P.Z., Davalos, J.F., and Barbero, E.J. 1994. “FRPBEAM: A Computer Program for

Analysis and Design of FRP Beams,” CFC-94-191, Constructed Facilities Center, West

Virginia University, Morgantown, WV. 80 pages.

Qiao, P.Z., Davalos, J.F., Barbero, E.J., and Troutman, D. 1999. "Equations Facilitate

Composite Designs," Modern Plastics Magazine, A publication of the McGraw-Hill

Companies, 76(11): 77-80. (Modern Plastics Magazine Award).

Qiao, P.Z, Davalos, J.F., and Wang, J.L. 2001. "Local buckling of composite FRP shapes by

discrete plate analysis," Journal of Structural Engineering, ASCE, 127 (3): 245-255.

Qiao, P.Z. and Zou, G.P. 2003. “Local Buckling of Composite Fiber-Reinforced Plastic Wide-

Flange Sections,” Journal of Structural Engineering, ASCE, 129(1): 125-129.

Qiao, P.Z. and Zou, G.P. (2002). “Local Buckling of Elastically Restrained Fiber-Reinforced

Plastic Plates and its Applications to Box-Sections,” Journal of Engineering Mechanics,

ASCE, 128 (12): 1324-1330.

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